with a transition probability \(B(|u|, \hat{u} \cdot \sigma )= \Phi (|u|) b(\hat{u} \cdot \sigma )\)  with the potential function \(\Phi \)  satisfying conditions from (REF ) and \(b\in L^{1}(\mathbb {S}^{d-1})\) . Such estimates are central when implementing an energy method for the Boltzmann equation. As stated earlier, these ideas started in [1]}, [2]} and were successfully concluded in [3]}. They are inspired in older ideas used for the analysis of the linearized Boltzmann equation where the gain operator is broken down in a big compact (smoothing) part that satisfies a gain of integrability property and a small rough part. Then, one uses the loss part of the collision operator to control such small rough part. In the following argument, we use Theorem REF  which reduces considerably the technicalities of the original arguments given in [3]}.
